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Doc: move info for btree opclass implementors into main documentation.

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Up to now, useful info for writing a new btree opclass has been buried
in the backend's nbtree/README file.  Let's move it into the SGML docs,
in preparation for extending it with info about "in_range" functions
in the upcoming window RANGE patch.

To do this, I chose to create a new chapter for btree indexes in Part VII
(Internals), parallel to the chapters that exist for the newer index AMs.
This is a pretty short chapter as-is.  At some point somebody might care
to flesh it out with more detail about btree internals, but that is
beyond the scope of my ambition for today.

Discussion: https://postgr.es/m/23141.1517874668@sss.pgh.pa.us
This commit is contained in:
Tom Lane
2018-02-06 13:52:27 -05:00
parent f069c91a57
commit 3785f7eee3
5 changed files with 276 additions and 61 deletions
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doc/src/sgml/btree.sgml Normal file
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<!-- doc/src/sgml/btree.sgml -->
<chapter id="btree">
<title>B-Tree Indexes</title>
<indexterm>
<primary>index</primary>
<secondary>B-Tree</secondary>
</indexterm>
<sect1 id="btree-intro">
<title>Introduction</title>
<para>
<productname>PostgreSQL</productname> includes an implementation of the
standard <acronym>btree</acronym> (multi-way binary tree) index data
structure. Any data type that can be sorted into a well-defined linear
order can be indexed by a btree index. The only limitation is that an
index entry cannot exceed approximately one-third of a page (after TOAST
compression, if applicable).
</para>
<para>
Because each btree operator class imposes a sort order on its data type,
btree operator classes (or, really, operator families) have come to be
used as <productname>PostgreSQL</productname>'s general representation
and understanding of sorting semantics. Therefore, they've acquired
some features that go beyond what would be needed just to support btree
indexes, and parts of the system that are quite distant from the
btree AM make use of them.
</para>
</sect1>
<sect1 id="btree-behavior">
<title>Behavior of B-Tree Operator Classes</title>
<para>
As shown in <xref linkend="xindex-btree-strat-table"/>, a btree operator
class must provide five comparison operators,
<literal>&lt;</literal>,
<literal>&lt;=</literal>,
<literal>=</literal>,
<literal>&gt;=</literal> and
<literal>&gt;</literal>.
One might expect that <literal>&lt;&gt;</literal> should also be part of
the operator class, but it is not, because it would almost never be
useful to use a <literal>&lt;&gt;</literal> WHERE clause in an index
search. (For some purposes, the planner treats <literal>&lt;&gt;</literal>
as associated with a btree operator class; but it finds that operator via
the <literal>=</literal> operator's negator link, rather than
from <structname>pg_amop</structname>.)
</para>
<para>
When several data types share near-identical sorting semantics, their
operator classes can be grouped into an operator family. Doing so is
advantageous because it allows the planner to make deductions about
cross-type comparisons. Each operator class within the family should
contain the single-type operators (and associated support functions)
for its input data type, while cross-type comparison operators and
support functions are <quote>loose</quote> in the family. It is
recommendable that a complete set of cross-type operators be included
in the family, thus ensuring that the planner can represent any
comparison conditions that it deduces from transitivity.
</para>
<para>
There are some basic assumptions that a btree operator family must
satisfy:
</para>
<itemizedlist>
<listitem>
<para>
An <literal>=</literal> operator must be an equivalence relation; that
is, for all non-null values <replaceable>A</replaceable>,
<replaceable>B</replaceable>, <replaceable>C</replaceable> of the
data type:
<itemizedlist>
<listitem>
<para>
<replaceable>A</replaceable> <literal>=</literal>
<replaceable>A</replaceable> is true
(<firstterm>reflexive law</firstterm>)
</para>
</listitem>
<listitem>
<para>
if <replaceable>A</replaceable> <literal>=</literal>
<replaceable>B</replaceable>,
then <replaceable>B</replaceable> <literal>=</literal>
<replaceable>A</replaceable>
(<firstterm>symmetric law</firstterm>)
</para>
</listitem>
<listitem>
<para>
if <replaceable>A</replaceable> <literal>=</literal>
<replaceable>B</replaceable> and <replaceable>B</replaceable>
<literal>=</literal> <replaceable>C</replaceable>,
then <replaceable>A</replaceable> <literal>=</literal>
<replaceable>C</replaceable>
(<firstterm>transitive law</firstterm>)
</para>
</listitem>
</itemizedlist>
</para>
</listitem>
<listitem>
<para>
A <literal>&lt;</literal> operator must be a strong ordering relation;
that is, for all non-null values <replaceable>A</replaceable>,
<replaceable>B</replaceable>, <replaceable>C</replaceable>:
<itemizedlist>
<listitem>
<para>
<replaceable>A</replaceable> <literal>&lt;</literal>
<replaceable>A</replaceable> is false
(<firstterm>irreflexive law</firstterm>)
</para>
</listitem>
<listitem>
<para>
if <replaceable>A</replaceable> <literal>&lt;</literal>
<replaceable>B</replaceable>
and <replaceable>B</replaceable> <literal>&lt;</literal>
<replaceable>C</replaceable>,
then <replaceable>A</replaceable> <literal>&lt;</literal>
<replaceable>C</replaceable>
(<firstterm>transitive law</firstterm>)
</para>
</listitem>
</itemizedlist>
</para>
</listitem>
<listitem>
<para>
Furthermore, the ordering is total; that is, for all non-null
values <replaceable>A</replaceable>, <replaceable>B</replaceable>:
<itemizedlist>
<listitem>
<para>
exactly one of <replaceable>A</replaceable> <literal>&lt;</literal>
<replaceable>B</replaceable>, <replaceable>A</replaceable>
<literal>=</literal> <replaceable>B</replaceable>, and
<replaceable>B</replaceable> <literal>&lt;</literal>
<replaceable>A</replaceable> is true
(<firstterm>trichotomy law</firstterm>)
</para>
</listitem>
</itemizedlist>
(The trichotomy law justifies the definition of the comparison support
function, of course.)
</para>
</listitem>
</itemizedlist>
<para>
The other three operators are defined in terms of <literal>=</literal>
and <literal>&lt;</literal> in the obvious way, and must act consistently
with them.
</para>
<para>
For an operator family supporting multiple data types, the above laws must
hold when <replaceable>A</replaceable>, <replaceable>B</replaceable>,
<replaceable>C</replaceable> are taken from any data types in the family.
The transitive laws are the trickiest to ensure, as in cross-type
situations they represent statements that the behaviors of two or three
different operators are consistent.
As an example, it would not work to put <type>float8</type>
and <type>numeric</type> into the same operator family, at least not with
the current semantics that <type>numeric</type> values are converted
to <type>float8</type> for comparison to a <type>float8</type>. Because
of the limited accuracy of <type>float8</type>, this means there are
distinct <type>numeric</type> values that will compare equal to the
same <type>float8</type> value, and thus the transitive law would fail.
</para>
<para>
Another requirement for a multiple-data-type family is that any implicit
or binary-coercion casts that are defined between data types included in
the operator family must not change the associated sort ordering.
</para>
<para>
It should be fairly clear why a btree index requires these laws to hold
within a single data type: without them there is no ordering to arrange
the keys with. Also, index searches using a comparison key of a
different data type require comparisons to behave sanely across two
data types. The extensions to three or more data types within a family
are not strictly required by the btree index mechanism itself, but the
planner relies on them for optimization purposes.
</para>
</sect1>
<sect1 id="btree-support-funcs">
<title>B-Tree Support Functions</title>
<para>
As shown in <xref linkend="xindex-btree-support-table"/>, btree defines
one required and one optional support function.
</para>
<para>
For each combination of data types that a btree operator family provides
comparison operators for, it must provide a comparison support function,
registered in <structname>pg_amproc</structname> with support function
number 1 and
<structfield>amproclefttype</structfield>/<structfield>amprocrighttype</structfield>
equal to the left and right data types for the comparison (i.e., the
same data types that the matching operators are registered with
in <structname>pg_amop</structname>).
The comparison function must take two non-null values
<replaceable>A</replaceable> and <replaceable>B</replaceable> and
return an <type>int32</type> value that
is <literal>&lt;</literal> <literal>0</literal>, <literal>0</literal>,
or <literal>&gt;</literal> <literal>0</literal>
when <replaceable>A</replaceable> <literal>&lt;</literal>
<replaceable>B</replaceable>, <replaceable>A</replaceable>
<literal>=</literal> <replaceable>B</replaceable>,
or <replaceable>A</replaceable> <literal>&gt;</literal>
<replaceable>B</replaceable>, respectively. The function must not
return <literal>INT_MIN</literal> for the <replaceable>A</replaceable>
<literal>&lt;</literal> <replaceable>B</replaceable> case,
since the value may be negated before being tested for sign. A null
result is disallowed, too.
See <filename>src/backend/access/nbtree/nbtcompare.c</filename> for
examples.
</para>
<para>
If the compared values are of a collatable data type, the appropriate
collation OID will be passed to the comparison support function, using
the standard <function>PG_GET_COLLATION()</function> mechanism.
</para>
<para>
Optionally, a btree operator family may provide <firstterm>sort
support</firstterm> function(s), registered under support function number
2. These functions allow implementing comparisons for sorting purposes
in a more efficient way than naively calling the comparison support
function. The APIs involved in this are defined in
<filename>src/include/utils/sortsupport.h</filename>.
</para>
</sect1>
<sect1 id="btree-implementation">
<title>Implementation</title>
<para>
An introduction to the btree index implementation can be found in
<filename>src/backend/access/nbtree/README</filename>.
</para>
</sect1>
</chapter>

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doc/src/sgml/filelist.sgml
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@ -83,6 +83,7 @@
<!ENTITY bki SYSTEM "bki.sgml"> <!ENTITY bki SYSTEM "bki.sgml">
<!ENTITY catalogs SYSTEM "catalogs.sgml"> <!ENTITY catalogs SYSTEM "catalogs.sgml">
<!ENTITY geqo SYSTEM "geqo.sgml"> <!ENTITY geqo SYSTEM "geqo.sgml">
<!ENTITY btree SYSTEM "btree.sgml">
<!ENTITY gist SYSTEM "gist.sgml"> <!ENTITY gist SYSTEM "gist.sgml">
<!ENTITY spgist SYSTEM "spgist.sgml"> <!ENTITY spgist SYSTEM "spgist.sgml">
<!ENTITY gin SYSTEM "gin.sgml"> <!ENTITY gin SYSTEM "gin.sgml">

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doc/src/sgml/postgres.sgml
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@ -252,6 +252,7 @@
&geqo; &geqo;
&indexam; &indexam;
&generic-wal; &generic-wal;
&btree;
&gist; &gist;
&spgist; &spgist;
&gin; &gin;

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doc/src/sgml/xindex.sgml
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@ -35,7 +35,7 @@
<productname>PostgreSQL</productname>, but all index methods are <productname>PostgreSQL</productname>, but all index methods are
described in <classname>pg_am</classname>. It is possible to add a described in <classname>pg_am</classname>. It is possible to add a
new index access method by writing the necessary code and new index access method by writing the necessary code and
then creating a row in <classname>pg_am</classname> &mdash; but that is then creating an entry in <classname>pg_am</classname> &mdash; but that is
beyond the scope of this chapter (see <xref linkend="indexam"/>). beyond the scope of this chapter (see <xref linkend="indexam"/>).
</para> </para>
@ -404,6 +404,8 @@
B-trees require a single support function, and allow a second one to be B-trees require a single support function, and allow a second one to be
supplied at the operator class author's option, as shown in <xref supplied at the operator class author's option, as shown in <xref
linkend="xindex-btree-support-table"/>. linkend="xindex-btree-support-table"/>.
The requirements for these support functions are explained further in
<xref linkend="btree-support-funcs"/>.
</para> </para>
<table tocentry="1" id="xindex-btree-support-table"> <table tocentry="1" id="xindex-btree-support-table">
@ -426,8 +428,8 @@
</row> </row>
<row> <row>
<entry> <entry>
Return the addresses of C-callable sort support function(s), Return the addresses of C-callable sort support function(s)
as documented in <filename>utils/sortsupport.h</filename> (optional) (optional)
</entry> </entry>
<entry>2</entry> <entry>2</entry>
</row> </row>
@ -1056,11 +1058,8 @@ ALTER OPERATOR FAMILY integer_ops USING btree ADD
<para> <para>
In a B-tree operator family, all the operators in the family must sort In a B-tree operator family, all the operators in the family must sort
compatibly, meaning that the transitive laws hold across all the data types compatibly, as is specified in detail in <xref linkend="btree-behavior"/>.
supported by the family: <quote>if A = B and B = C, then A = C</quote>, For each
and <quote>if A &lt; B and B &lt; C, then A &lt; C</quote>. Moreover, implicit
or binary coercion casts between types represented in the operator family
must not change the associated sort ordering. For each
operator in the family there must be a support function having the same operator in the family there must be a support function having the same
two input data types as the operator. It is recommended that a family be two input data types as the operator. It is recommended that a family be
complete, i.e., for each combination of data types, all operators are complete, i.e., for each combination of data types, all operators are

53
src/backend/access/nbtree/README
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@ -623,56 +623,3 @@ routines must treat it accordingly. The actual key stored in the
item is irrelevant, and need not be stored at all. This arrangement item is irrelevant, and need not be stored at all. This arrangement
corresponds to the fact that an L&Y non-leaf page has one more pointer corresponds to the fact that an L&Y non-leaf page has one more pointer
than key. than key.
Notes to Operator Class Implementors
------------------------------------
With this implementation, we require each supported combination of
datatypes to supply us with a comparison procedure via pg_amproc.
This procedure must take two nonnull values A and B and return an int32 < 0,
0, or > 0 if A < B, A = B, or A > B, respectively. The procedure must
not return INT_MIN for "A < B", since the value may be negated before
being tested for sign. A null result is disallowed, too. See nbtcompare.c
for examples.
There are some basic assumptions that a btree operator family must satisfy:
An = operator must be an equivalence relation; that is, for all non-null
values A,B,C of the datatype:
A = A is true reflexive law
if A = B, then B = A symmetric law
if A = B and B = C, then A = C transitive law
A < operator must be a strong ordering relation; that is, for all non-null
values A,B,C:
A < A is false irreflexive law
if A < B and B < C, then A < C transitive law
Furthermore, the ordering is total; that is, for all non-null values A,B:
exactly one of A < B, A = B, and B < A is true trichotomy law
(The trichotomy law justifies the definition of the comparison support
procedure, of course.)
The other three operators are defined in terms of these two in the obvious way,
and must act consistently with them.
For an operator family supporting multiple datatypes, the above laws must hold
when A,B,C are taken from any datatypes in the family. The transitive laws
are the trickiest to ensure, as in cross-type situations they represent
statements that the behaviors of two or three different operators are
consistent. As an example, it would not work to put float8 and numeric into
an opfamily, at least not with the current semantics that numerics are
converted to float8 for comparison to a float8. Because of the limited
accuracy of float8, this means there are distinct numeric values that will
compare equal to the same float8 value, and thus the transitive law fails.
It should be fairly clear why a btree index requires these laws to hold within
a single datatype: without them there is no ordering to arrange the keys with.
Also, index searches using a key of a different datatype require comparisons
to behave sanely across two datatypes. The extensions to three or more
datatypes within a family are not strictly required by the btree index
mechanism itself, but the planner relies on them for optimization purposes.